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Chapter 5

Time Value of Money

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Figure 5.2 Compounding and Discounting

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Figure 5.3 Calculator Keys

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Future Value of a Single Amount

• Future value is the value at a given future date of an amount placed on deposit today and earning interest at a specified rate. Found by applying compound interest over a specified period of time.

• Compound interest is interest that is earned on a given deposit and has become part of the principal at the end of a specified period.

• Principal is the amount of money on which interest is paid.

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Future Value of a Single Amount: The Equation for Future Value

• We use the following notation for the various inputs:

– FVn = future value at the end of period n

– PV = initial principal, or present value

– r = annual rate of interest paid. (Note: On financial calculators, I is typically used to represent this rate.)

– n = number of periods (typically years) that the money is left on deposit

• The general equation for the future value at the end of period n is

FVn = PV (1 + r)n

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Future Value of a Single Amount: The Equation for Future Value

Jane Farber places $800 in a savings account paying 6% interest compounded annually. She wants to know how much money will be in the account at the end of five years.

This analysis can be depicted on a time line as follows:

FV5 = $800 (1 + 0.06)5 = $800 (1.33823) = $1,070.58

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Present Value of a Single Amount

• Present value is the current dollar value of a future amount—the amount of money that would have to be invested today at a given interest rate over a specified period to equal the future amount.

• It is based on the idea that a dollar today is worth more than a dollar tomorrow.

• Discounting cash flows is the process of finding present values; the inverse of compounding interest.

• The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, or the cost of capital.

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Personal Finance Example

Paul Shorter has an opportunity to receive $300 one year from now. If he can earn 6% on his investments, what is the most he should pay now for this opportunity?

PV (1 + 0.06) = $300

PV = $300/(1 + 0.06) = $283.02

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Present Value of a Single Amount: The Equation for Present Value

The present value, PV, of some future amount, FVn, to be received n periods from now, assuming an interest rate (or opportunity cost) of r, is calculated as follows:

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Present Value of a Single Amount: The Equation for Future Value

Pam Valenti wishes to find the present value of $1,700 that will be received 8 years from now. Pam’s opportunity cost is 8%.

This analysis can be depicted on a time line as follows:

PV = $1,700/(1 + 0.08)8 = $1,700/1.85093 = $918.46

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Figure 5.5 Present Value Relationship